SD=2
Plot[{PDF[NormalDistribution[0,SD],x]},{x,-5,5}]

Plot[{PDF[StudentTDistribution[0,SD,100],x]},{x,-5,5},Filling->Axis]

Plot[{PDF[StudentTDistribution[0,SD,3],x],PDF[NormalDistribution[0,SD],x]},{x,-5,5},Filling->Axis]

SD

Integrate[{((E^(-((x^2)/(2*(SD^2)))))/(SD*(Sqrt[2 π])))},{x,-10,10}]
N[%]

yRange=1

Integrate[{((E^(-((x^2)/2)))/(Sqrt[2 π]))},{x,-yRange,yRange}]
N[%]

Plot[{PDF[NormalDistribution[0,1.0],x],
PDF[NormalDistribution[3,1],x]},{x,-5,5}]

Plot[{PDF[NormalDistribution[0,1],x],
PDF[NormalDistribution[3,1],x]},{x,-5,5},Filling->Axis]

Plot[PDF[NormalDistribution[0,1],x]*
PDF[NormalDistribution[3,1],x],{x,-5,5},Filling->Axis]

MeanSeparation=2

Integrate[PDF[NormalDistribution[0,1],x]*
PDF[NormalDistribution[MeanSeparation,1],x],{x,-100,100}]

N[%]/2

`